Edge-arc-disjoint paths in semicomplete mixed graphs

Abstract

The so-called weak-2-linkage problem asks for a given digraph $D=\left(V,A\right)$ and distinct vertices $s_1,s_2,t_1,t_2$ of $D$ whether $D$ has arc-disjoint paths $P_1,P_2$ so that $P_i$ is an $\left(s_i,t_i\right)$-path for $i=1,2$. This problem is NP-complete for general digraphs but the first author showed that the problem is polynomially solvable and that all exceptions can be characterized when $D$ is a semicomplete digraph, that is, a digraph with no pair of nonadjacent vertices. In this paper we extend these results to paths which are both edge-disjoint and arc-disjoint in semicomplete mixed graphs, that is, a mixed graph $M=\left(V,E\cup A\right)$ in which every pair of distinct vertices has either an arc, an edge, or both an arc and an edge between them. We give a complete characterization of the negative instances and explain how this gives rise to a polynomial algorithm for the problem.